Mass and energy · Capstone
Rebuild the September argument
Which premise does the work, and which subtraction removes what cannot be measured?
What to do with this page
Explain to someone else why a body that gives off light loses mass, which subtraction removes what cannot be measured, and why the finite-speed quotient is not the exact mass loss.
Each claim below links to the passage it is read from. Follow the links and the argument is the paper's; read only this page and it is a summary of the paper, which is a different thing and says so.
The six claims, in the order the paper makes them
The chain below fixes what must come before what, and 2 arrangements satisfy it. The paper prints one of them; the others are not mistakes.
An imported resultNeeds nothing before it
The energy of a light complex transforms between two frames by a rule the June paper on the electrodynamics of moving bodies established in its section eight. This paper quotes that result instead of deriving it again.
The imported result is the one step this paper does not take for itself. Its provenance is the June paper, quoted here in the opening.
An assumptionNeeds nothing before it
A body at rest gives off equal amounts of light in opposite directions, so it is still at rest in the frame where it began.
A derivationUses claim 1 and claim 2
Write the body's energy before and after the emission in both frames. Each frame gives a balance, and in the moving frame the two pulse energies depend on the emission angle while their sum does not.
A derivationUses claim 3
Subtracting one balance from the other removes the internal energies nobody has measured, provided the difference between a body's two frame energies is its kinetic energy plus a constant that the emission leaves unchanged.
The cancellation does not establish the premise it leans on. The paper asserts that the constant is the same before and after the emission, and later writers questioned that.
A derivationUses claim 4
Where the speed is small against the speed of light, the surviving difference takes the Newtonian form, with a coefficient carrying the emitted energy divided by the square of the speed of light.
The low-speed form is an approximation. Treating its quotient as the exact mass loss is the mistake this capstone is built around.
A generalizationUses claim 5
The body's mass after the emission is smaller by the emitted energy divided by the square of the speed of light, and the paper then states the general proposition that a body's mass measures its energy content.
The general proposition reaches past radiation to any energy a body gives up. The paper states it; the argument above establishes the radiation case.
What the argument is granted
Every claim above names the assumptions it uses. These are the things the paper is given or asserts rather than establishes, and the fourth claim is where one of them does the work.
Premise
The light-energy transformation imported from the earlier paper's section eight, whose provenance this paper states in its own opening.
Premise
The difference between a body's energy in the moving frame and in its rest frame is its kinetic energy plus a constant, and that constant is the same before and after the emission. The paper asserts this rather than deriving it.
Approximation
The Newtonian form of kinetic energy, which holds where the speed is small against the speed of light.
Setup
The two emissions are equal and opposite, so the body does not recoil.
Choice of boundary
The system under discussion is the body alone, not the body together with the radiation it has sent away.
The displays this argument turns on
Two opposite pulses add up to gamma L
How the two pulse energies divide in the moving frame, and what their sum does not depend on.
L over two gamma times one minus beta cos phi, plus L over two gamma times one plus beta cos phi, equals gamma L.
Subtract the two ledgers
The subtraction that removes the internal energies.
H zero minus E zero, minus H one minus E one, equals gamma L minus L, which equals L times gamma minus one.
At low speed, the drop looks like a kinetic energy
What the surviving difference becomes once the speed is small.
K zero minus K one is approximately one half times L over c squared times v squared.
Where to watch the quantities move
Watch the two pulse energies change with the emission angle while their sum stays where it is.
Follow the exact difference beside its low-speed proxy as the speed falls.
What this does not claim
The paper never writes the equation it is now quoted for, and this capstone does not supply it. Nothing here says that mass turns into light: the body loses inertia in proportion to the energy it gives off. No ledger on this page carries an absolute rest energy.